Lifting of homeomorphisms to 4-sheeted branched coverings of a disk
Let \(p:X\to D\) be a 4-sheeted branched covering of a closed disk \(D\), possibly not connected, with \(n\) branch values \(A_1\), \(A_2,\ldots,A_n\in D^\circ\). It is supposed that \(p\) is simple, i.e. for each \(A_j\) the preimage \(p^{-1}(A_j)\) consists of \(3\) points. Each loop \(\sigma\) at a fixed point \(A_0\in\partial D\), lying in \(D\setminus\{A_1\), \(A_2,\ldots,A_n\}\), defines a permutaion of leaves of the branched covering, so we have the monodromy homeomorphism \(\mu:\pi_1(D\setminus\{A_1\), \(A_2,\ldots,A_n\},A_0)\to\Sigma_4\); here \(\Sigma_4\) is the symmetric group of order~\(4\). We fix a sequence of \(n\) disjoint (except for the initial points) simple curves \(\alpha_1\), \(\alpha_2,\ldots,\alpha_n\); each \(\alpha_j\) connects \(A_0\) to \(A_j\) in \(D\) and does not meet \(A_i\), \(i\neq j\). Let the curves \(\alpha_1\), \(\alpha_2,\ldots,\alpha_n\) meet at \(A_0\) in the clockwise order. For each \(j\) denote by \(\widehat{\alpha}_j\) the loop which goes along \(\alpha_j\) to a sufficiently small neighbourhood \(U_j\) of \(A_j\) then turns clockwise around \(A_j\) in \(U_j\) and comes back to \(A_0\) along \(\alpha_j\). Denote by \(\mu(\alpha_j)\) the image of the homotopy class of \(\widehat{\alpha}_j\) under the monodromy homeomorphism. By definition, the sequence \(\mu(\alpha_1)\), \(\mu(\alpha_2),\dots,\mu(\alpha_n)\) is a monodromy sequence for \(p\). Now we remember the concept of half-twist. Let \(x\) be a simple arc in \(D^\circ\) connecting two distinct branch values and free of the others. Consider a sufficiently small neighbourhood of \(x\). It is homeomorphic to the closed unit disk \(U\) with \(x\) corresponding to a segment centered at the origin. The half-twist around \(x\) is the isotopy class of the homeomorphism induced by the following homeomorphism \(h\) of \(U\). The map \(h\) is obtained from the identity map by rotation of \(y\) around the origin by \(180^\circ\) with \(\partial U\) pointwise fixed. The isotopy classes of homeomorphisms of \(D\) which are fixed on \(\partial D\) and permute the branch values form the braid group \(B_n\). Consider the classes of homeomorphisms which can be lifted to homeomorphisms of \(X\). They form a subgroup \(L(p)\) in \(B_n\). The main goal of the paper is to describe explicitly generators for \(L(p)\) for each equivalence class of coverings. The authors describe \(8\) possible types of different monodromy sequences. For each type they construct explicitly so-called distinguished arcs. Theorem~3 (Main Theorem). Let \(p:X\to D\) be a simple, 4-sheeted branched covering of a disk \(D\) with \(n\) branch points, determined by a monodromy sequence of type~\(j\), \(1\leq j\leq 8\). Then the group \(L(p)\) of isotopy classes of the liftable homeomorphisms of \(D\) is generated by the liftable powers of half-twists with respect to the distinguished arcs described for type~\(j\). The liftability of powers of half-twists can be easily verified using the structure of the preimage of the corresponding arcs.
scientific article; zbMATH DE number 5346285